To obtain a Lorentz invariant Schrödinger equation, we considered the square root of the Klein–Gordon equation. This had the disadvantage that the Hamiltonian https://www.w3.org/1998/Math/MathML"> H = p → 2 + m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_603.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> contains an infinite number of powers of https://www.w3.org/1998/Math/MathML"> p → 2 / m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_604.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , the parameter in which the square root should be expanded. It would have been better to treat space and time on a more equal footing in the Schrödinger equation. This is what Dirac took as his starting point. As the Schrödinger equation is linear in https://www.w3.org/1998/Math/MathML"> p 0 = i ∂ / ∂ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_605.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , one is looking for a Hamiltonian that is linear in the momenta https://www.w3.org/1998/Math/MathML"> p j = i ∂ / ∂ x j ( = − p j ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_606.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> ). ( 12.1 ) https://www.w3.org/1998/Math/MathML"> i ∂ Ψ ∂ t = H Ψ = − i α k ∂ Ψ ∂ x k + β m Ψ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/03ca065d-1bf0-4218-811d-2f95276252a3/content/equ_607.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
We re-examine the square of the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , in Equation 15.7 and also the Dirac approach in finding the relativistic Hamiltonian. The classical Hamiltonian for the free particle should be the positive square root of Equation 15.7 which is not a linear function of the momenta. Applying this to the problem, say for example in Equation 15.10, we have the resultant wave equation that is asymmetrical with respect to space and time derivatives and so the necessity to linearize the equation in the momentum operators. So for Equation 15.10, it is impossible to include the Pauli spin matrices without destroying the invariance of the theory since the spin matrices transform the components of a three-dimensional rather than a four-dimensional vector. In addition, the function, https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , has one component rather than two components like spinors and so the Schrödinger relativistic equation represents a particle with no spin. So for a potential to be added in Equation 15.10, the Lorentz-transformation properties of such an addition should be guaranteed by matrix algebra which is the requirement for the Hamiltonian https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to be linear in the momentum operators and guarantees relativistic invariance. This matrix algebra which permits the linearization of the Hamiltonian in the momentum operator was introduced by Dirac with the help of the Pauli matrices.
We look for a Hamiltonian to use in the time-dependent Schrödinger equation, and because this is first order in time we stay first order in space trying to treat all coordinates on a similar basis. Write 4.1 https://www.w3.org/1998/Math/MathML"> i ∂ ψ ∂ t = H ψ = − i α i ∂ ψ ∂ x i + m β ψ i = 1 , 2 , 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/8ac61231-bcf3-4e9e-a16f-9d6f78005bbb/content/math4_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> where ψ is now a column and the α i and β are square matrices.
Before examining the Dirac equation that gives a relativistic quantum mechanical description of the electron, we will examine the Klein-Gordon equation. Compared to the Schrödinger equation, the Dirac equation should have an appreciable modification in the quantum description of the properties of the electron. In the non-relativistic quantum mechanical formulation, we begin with the classical relation for the energy in the absence of interaction, i.e., the kinetic energy: 15.1 https://www.w3.org/1998/Math/MathML"> Η = p 2 2 m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math15_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
In the previous chapter, we did not examine the change of a state with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . It has been assumed that we have been examining a state at a given moment https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In the general case, the state of a system changes with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . For example, say the motion of an electron in a potential field, its wave function depends on time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : https://www.w3.org/1998/Math/MathML"> Ψ = Ψ q , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Such a dependence has to be described by a differential equation in which we may have https://www.w3.org/1998/Math/MathML"> ∂ Ψ ∂ t , ∂ 2 Ψ ∂ t 2 , ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> .
Consider the properties of a harmonic oscillator: The motion is bounded: At infinity, the wave function https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> tends to zero The energy levels are discrete These energy states are non-degenerate, i.e., any energy level corresponds to one eigenfunction The oscillatory theorem: The wave function https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> of the https://www.w3.org/1998/Math/MathML"> n th https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> state has https://www.w3.org/1998/Math/MathML"> n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> zeros but not considering infinite far removed points